Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on R2

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem {Delta u + u = (I-alpha * vertical bar v vertical bar(p)) v(p-1) in R-N, -Delta v + v = (I-beta * vertical bar u vertical bar(q))u(q-1) in R-N, u(x),v(x) -> 0 when vertical bar x vertical bar -> infinity, when (N + alpha)/p,+ (N + beta)/q <= 2(N - 2) (if N >= 3) and (N + alpha)/p (N + beta)/q >= 2N (if N = 2), where I-alpha and I-beta denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem { Delta u + u = (I-alpha * vertical bar v vertical bar(alpha/2+1))vertical bar v vertical bar(alpha/2-1) v + g(v) in R-2, -Delta u + v = (I-beta * vertical bar u&VERBAR(;beta/2+1))vertical bar u vertical bar(beta/2-1) u + f(u), in R-2, u, v is an element of H-1 (R-2), where alpha, beta (0, 2) and f, g have exponential critical growth in the Trudinger-Moser sense.